Rev. J. Challis on the Theory of the Ball-Pendulum. 185 



since the determination of the minimum depends not alone 

 upon the choice of the signs -f or — in the equation (7), but 

 also upon the values of \ p, and p lt . The way in which this 

 takes place is too obvious to need any further remark. 



The interpretation, therefore, of Bernoulli's equation is 

 completed and the problem at the same time resolved in the 

 most general case that can be proposed on a spherical earth, 

 and with a uniform declination for the period of twilight on 

 the day in question. It might hence seem that the inquiry, 

 in reference to everything of value which it can afford, might 

 be properly concluded here. Nevertheless, as the curious pro- 

 perties which Delambre has deduced, both in his Astronomic 

 and in his Histoire de V Astronomic^ for the particular case he 

 considered, have in several instances remarkable analogies in 

 the more general one above discussed, it will not be out of 

 place to annex a few of them to the preceding investigation. 

 [To be continued.] 



XXXIV. Theory of the Correction to be applied to a Ball- 

 Pendulum for the Reduction to a Vacuum. By the Rev. 

 J. Challis, Fellow of the Cambridge Philosophical Society*. 



TN a previous communication respecting the resistance to 

 *- the motion of small spherical bodies in elastic fluids f, I at- 

 tempted to explain, entirely from theoretical considerations, 

 the manner inwhich the air is acted upon, when a pendulum 

 consisting of a small spherical ball suspended by a very slen- 

 der wire, performs very small oscillations in it ; but I omitted 

 to enter upon any calculation to ascertain the numerical value 

 of the correction required for reducing the time of vibration 

 in air to that in a vacuum. As the theory there advanced is 

 competent to obtain such a result without the aid of experi- 

 ment, I propose to make this the object of the present com- 

 munication. 



The following equation was obtained in the paper referred 

 to:— 



MdHkp 8 = 2g(M-i*) (h-z), 



in which M is the mass of the ball, v the velocity of its centre, 

 /* the mass of an equal volume of air, g the force of gravity, 

 h — x the vertical descent of the centre of the ball. The equa- 

 tion without the term mv% is that which was formerly em- 



* Communicated by the Author, 



f See Lond. and Edinb. Phil. Mag. vol. i. p. 40. 



Third Series. Vol. 3. No. 15. Sept. 1633. 2 B 



